Engineering Mat...

The Eigen values of the matrix \(\left[ {\begin{array}{*{20}{c}}3&0&0\\0&2&{ - 3}\\0&1&{ - 2}\end{array}} \right]\) are

2x + 3y + 5z = 9

7x + 3y – 2z = 8

2x + 3y + λz = 7

Find the value of λ for which the vectors given are linearly dependent:

\(\begin{array}{l} \vec a = i + 2\hat j + 3\hat k\\ \vec b = 4i + \lambda \hat j + 6\hat k\\ \vec c = 3i + 4\hat j + 5\hat k \end{array}\)

The number of distinct real values of x for which the matrix \(\left( {\begin{array}{*{20}{c}} x&1&1\\ 1&x&1\\ 1&1&x \end{array}} \right)\) is singular is

If \(A = \left[ {\begin{array}{*{20}{c}}1&1&3\\5&2&6\\{ - 2}&{ - 1}&{ - 3}\end{array}} \right]\), then the value of A¹⁴ + 3A – 2I is

A 2 × 2 matrix is given as \(\left[ {\begin{array}{*{20}{c}} 1&p\\ q&2 \end{array}} \right].\) If the eigen values of the matrix are real and positive, then which one of the following relations should be satisfied?

Consider a Matrix \(M = {u^T}{v^T}\) where \(u = \left( {1\;1\;2} \right)\;and\;v = \;\left( {\begin{array}{*{20}{c}} 1\\ 2\\ 1 \end{array}} \right)\) also \({u^T}\) denotes the transpose of matrix u. Find the largest eigenvalue of M?

The value of α for which the system of equations

x – y – 3z = 3

2x + z = 0

-2y -7z = α

Consider the 3 × 3 matrix

\(A=\left( \begin{matrix} 0 & -1 & -1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \\\end{matrix} \right)\)